Counting differently: number systems and the bodies behind them

Most languages count in tens, and the reason is not mathematical elegance. It is that people have ten fingers.

Which immediately suggests the alternatives, and those alternatives exist.

Base twenty

Counting in twenties, using fingers and toes, is documented across a wide range of languages.

Traces survive in European languages where the main system is now decimal. French counts eighty as four twenties, and older forms of the language extended this further. Danish number words contain vestiges of a twenty-based system. Various Celtic languages use twenty-based counting for some numbers, and English retains the word score in older usage.

Full base-twenty systems have been used in Mesoamerican languages, in parts of West Africa, in the Caucasus and elsewhere. Maya mathematics used a base-twenty positional system with a symbol for zero, developed independently of the Old World systems.

Base twelve and base sixty

Twelve has practical advantages: it divides evenly by two, three, four and six, which makes trade and measurement easier than a base that divides only by two and five.

Duodecimal traces are everywhere in English measurement — twelve inches, twelve in a dozen, twelve months, and the fact that eleven and twelve have their own names rather than following the pattern of thirteen onward.

One proposed origin involves counting the finger bones of one hand with the thumb: three segments on each of four fingers gives twelve, and the other hand can count groups of twelve, giving sixty. This method is documented in use in parts of the Middle East and South Asia.

Base sixty comes from Mesopotamia, and it is the reason we have sixty seconds, sixty minutes and three hundred and sixty degrees. Those units have survived four thousand years and every subsequent attempt at metrication.

Body-part counting

A number of languages, particularly in New Guinea, use extended body-part tally systems, counting up one arm, across the head and down the other side, with each point having a name that also functions as a number.

These systems often have a defined limit — the number of points on the body — and larger quantities are handled differently, by repeating cycles or by other means.

Languages with few number words

Some languages have very restricted numeral systems — words for one, two and many, or similar.

This has generated a substantial and contested research literature, particularly around whether speakers of such languages can perform exact quantity tasks with larger numbers.

The findings that seem reasonably robust: speakers can perceive and compare quantities perfectly well, and exact matching of larger sets becomes harder without number words to track them. Speakers of such languages learn other counting systems without difficulty when they need to.

The strong claim that these speakers cannot conceive of quantity is not supported, and it has been used in ways that are worth being suspicious of.

Grammatical number beyond singular and plural

A separate matter, and a widespread one.

Many languages have a dual — a distinct form for exactly two — and some have a trial or a paucal for small numbers. Arabic, Slovene, and a range of Austronesian and Indigenous Australian languages have systems more elaborate than the English singular-plural distinction.

Classifier systems, found across East and Southeast Asia and elsewhere, require a word between the number and the noun that categorises the thing being counted — with different classifiers for flat things, long things, animals, people, machines and so on. Learners find these difficult and they are entirely systematic once learned.

Writing numbers

The positional decimal system with a zero, now used essentially worldwide, developed in India and reached Europe through Arabic-speaking intermediaries, which is why the digits are called Arabic numerals in English and Indian numerals in Arabic.

Its advantage over the alternatives is arithmetic. Adding and multiplying in Roman numerals is genuinely difficult, which is why calculation in medieval Europe was done on counting boards and abacuses rather than in writing.

Zero as a positional placeholder and as a number in its own right was the crucial component, and its adoption in Europe was slow — resisted for a period by commercial and municipal authorities partly on the grounds that the digits were easier to falsify in ledgers.

What the traces tell you

The irregular number words in a language are usually the oldest layer, preserving an earlier system.

English eleven and twelve derive from forms meaning roughly one left and two left — that is, after ten. The teens and the tens follow a different pattern. Score survives in older texts. Dozen and gross come from twelve. Each fragment is a piece of a system that was replaced and never completely erased.

Which is generally how language change works: the new system takes over, and the highest-frequency items resist longest, leaving a fossil record in the words people say most often.